A channel estimation process estimates a channel transfer function (CTF) at each subcarrier. The channel estimation is based on two special symbols called Channel Estimation (CE), which are part of a preamble of each packet.
A PHY preamble is attached to a beginning of every transmitted PHY payload. The preamble includes reference signals which can be useful to facilitate receivers in detecting and acquiring physical layer parameters required to properly decode the packet, such as gain, frequency-offset, timing information and channel estimation.
The PHY preamble usually has 3 consecutive sequences:                1. A Short Sequence, which has 0 or 12 repetitions of a Short-Symbol (SS), which can be useful to facilitate convergence of the receiver's AGC setting.        2. A Long Sequence (LS), which has 0, 4 or 8 repetitions of a Long-Symbol, which can be useful to facilitate recovery by the receiver of timing and frequency-offset information.        3. The Channel Estimation Sequence, which has 2 repetitions of a CE-Symbol and is useful to facilitate the receiver's estimation of Channel Transfer Function (CTF). The receiver equalizer coefficients are obtained from the inverse of the CTF coefficients.        
The arrival of a packet is detected via the LS of the preamble. Then, the CFO and possibly the Sampling Frequency Offset (SFO) are estimated using also the LS of the preamble and corrected. The CFO and SFO are generated due to the frequency offsets between the transmitter and the receiver local RF oscillators (LOs) and the sampling clock, respectively. The CFO and sometimes the SFO are estimated and corrected before the channel is estimated. However, the residual CFO may be too large causing degradation to the accuracy of the channel estimation, which may be significant for systems that use high constellations such as 1024-QAM and 4096-QAM as are used in MoCA and G.hn, respectively.
The residual CFO, which is the left-over CFO after estimating the CFO using the LS, affects the estimated channel transfer function (CTF) and the equalizer coefficients. This residual CFO can be estimated via the CE sequence. However, since the derived residual CFO (from the CE sequence) is obtained after the FFT of the CE sequence, it cannot be used to correct the CE before the FFT is applied for the use of the Channel Estimator/EQ block. In this work we show the effect of the residual CFO on the estimated equalizer coefficients and suggest how to use the estimated residual CFO based on the CE sequence to mitigate the affected estimated equalizer coefficients.
The received CE after FFT at a k-th tone of an n-th symbol is given by:Xn,k=Hn,k·sn,k+Nn,k,  Equation 1
where:
Hn,k is the frequency response of the k-th tone for symbol n,
sn,k is the signal modulating the k-th subcarrier (tone) of the n-th symbol and is known at the receiver,
Nn,k is additive white Gaussian noise (AWGN) at the k-th subcarrier of the n-th
symbol,
k is the frequency index from −N/2 to N/2−1,
N is the number of the FFT samples.
It's assumed that the channel Hn,k does not change from symbol to symbol within a packet and is a function of the subcarrier index k only. Since the signals modulating the subcarriers of the two CE symbols, sk, are the same, the received two CE symbols for n=1 and 2 are given by:X1,k=Hk·sk+N1,k X2,k=Hk·sk+N2,k  Equation 2
The channel estimation is obtained from the two identical training sequences called CE, which are part of the preamble. The standard channel estimation is done as follows:                Averaging of the two CE symbols after being converted to the frequency domainYk=(X1,kX2,k)/2  Equation 3        The least squares estimation of the channel frequency response per subcarrier is given by        
                              H          k          LS                =                              Y            k                                s            k                                              Equation        ⁢                                  ⁢        4                            The channel frequency response estimate can be improved by using the frequency domain correlation and the time domain correlation of the channel frequency response. The improved channel frequency response is generally a linear combination of the least squares channel coefficients, HkLS. As is shown later, the phase error of the channel coefficients due to the residual CFO is fixed per subcarrier; thus, the improvement of the channel estimate is not affected by the fixed phase error.        The equalizer coefficients, Qk, per each subcarrier k are estimated via        
                              Q          k                =                  1                      H            k            LS                                              Equation        ⁢                                  ⁢        5            
The residual CFO causes phase and gain errors on the channel estimation.
Assuming the phase difference between the two CE symbols due to the residual CFO is φ radians. Then, the received samples of the two CE symbols in the frequency domain are given by:X1,k=e−j·φ·Hk·sk+N1,k X2,k=Hk·sk+N2,k  Equation 6
Note that it is assumed that the phase of the 2nd CE symbol is 0 and the previous CE symbol is related to the second CE symbol. The channel estimation and the equalizer coefficients are related to the second CE symbol which is just before the arrival of the data symbols.
Averaging of the two CE symbols gives:Yk=(X1,k+X2,k)/2
After ignoring the noise:
            Y      k        =                  (                                            ⅇ                                                -                  j                                ·                ϕ                                      ·                          H              k                        ·                          s              k                                +                                    H              k                        ·                          s              k                                      )            /      2                  Y      k        =                            1          +                      ⅇ                                          -                j                            ·              ϕ                                      2            ·              H        k            ·              s        k            
The least squares estimate of the channel frequency response per subcarrier is given by:
                              H          k                      LS            ⁡                          (              1              )                                      =                              Y            k                                s            k                                              Equation        ⁢                                  ⁢        7                                          H          k                      LS            ⁡                          (              1              )                                      =                                            1              +                              ⅇ                                                      -                    j                                    ·                  ϕ                                                      2                    ·                      H            k                                              Equation        ⁢                                  ⁢        8            
Therefore, the channel estimation can be presented by:HkLS(1)=G·ej·Δφ·Hk 
where, the gain and phase distortions of the channel frequency response estimation are G and Δφ, respectively.
      H    k          LS      ⁡              (        1        )              =                                          ⅇ                                          -                j                            ·                              ϕ                /                2                                              ·                      (                                          ⅇ                                  j                  ·                                      ϕ                    /                    2                                                              +                              ⅇ                                                      -                    j                                    ·                                      ϕ                    /                    2                                                                        )                          2            ·              H        k              =                  ⅇ                              -            j                    ·                      ϕ            /            2                              ·              cos        ⁡                  (                      ϕ            /            2                    )                    ·              H        k            
The residual CFO causes a gain distortion of cos(φ/2) and phase distortion of −φ/2 on the channel estimation for every subcarrier. Note that the phase distortion of the channel transfer function estimate is only half of the residual CFO phase error in radian because the channel estimate averages the two CE symbols, where only the first one has the CFO phase error. The phase error of the estimated channel transfer function is fixed for all the frequency indexes.
The gain distortion can be approximated by
            cos      ⁡              (                  ϕ          /          2                )              ≈          1      -                        ϕ          2                8              ≈    1    ,          ⁢      ϕ    ⁢          <<      1      
The gain error of the channel transfer function due to the residual CFO phase error of one OFDM symbol is related to the square of the residual CFO phase error over one symbol while the phase distortion is related to the CFO phase. Therefore, the gain distortion is usually small and can be ignored. If the gain error is significant, a gain correction can be added.
The approximated channel transfer function estimation is given by:HkLS(1)≈e−j·φ/2·Hk 
The equalizer coefficients, Qk, per each subcarrier k are given by:
                                          Q            k                    =                      1                          H              k              LS                                      ⁢                                  ⁢                              Q            k                    ≈                                    ⅇ                              j                ·                                  ϕ                  /                  2                                                      ·                          1                              H                k                                                                        Equation        ⁢                                  ⁢        9            
The estimated equalizer coefficient per subcarrier has a constant phase error equals to half of the residual CFO phase error over one OFDM symbol.
Accordingly, a new apparatus and method are needed to mitigate CFO to achieve optimal receiver performance.